Quantum spin glasses require complex circuits for state preparation
Beyond Light Cones: State Preparation Complexity in Quantum Spin Glasses
Computational Complexity
Summary
Preparing the lowest-energy states of dense quantum spin glasses is very hard and needs complicated quantum circuits. The authors show that simple or shallow circuits cannot get close to these states, even if you have extra helper qubits. They introduce new mathematical methods to prove that many quantum gates and a lot of circuit depth are necessary. This helps us understand fundamental limits on how quantum computers can simulate such complex systems.
What this means in practice
- •For quantum hardware developers: Design quantum devices knowing that preparing low-energy states in dense spin glass models requires at least quadratic gate counts and certain circuit depths.
- •For quantum compiler engineers: Optimize compilers to recognize inherent lower bounds on circuit complexity when targeting ground states of dense quantum spin glasses, avoiding futile attempts at overly shallow circuits.
A theory result. No direct application yet.
Authors
Omar Al-Ghattas, David Gamarnik, Bobak T Kiani
Abstract
We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $Ω(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.